对任意的正实数
t
{\displaystyle t}
t
{\displaystyle t}
概率密度函数 是：
f
W
t
(
x
)
=
1
2
π
t
e
−
x
2
/
2
t
.
{\displaystyle f_{W_{t}}(x)={\frac {1}{\sqrt {2\pi t}}}e^{-x^{2}/{2t}}.}
这是因为按照维纳过程的定义，当
s
=
0
{\displaystyle s=0}
W
t
{\displaystyle W_{t}}
W
t
=
W
t
−
W
0
∼
N
(
0
,
t
)
.
{\displaystyle W_{t}=W_{t}-W_{0}\sim {\mathcal {N}}(0,t).}
它的数学期望是零：
E
(
W
t
)
=
0.
{\displaystyle \mathbb {E} (W_{t})=0.}
方差 是
t
{\displaystyle t}
Var
(
W
t
)
=
E
(
W
t
2
)
−
E
2
(
W
t
)
=
E
(
W
t
2
)
−
0
=
E
(
W
t
2
)
=
t
.
{\displaystyle \operatorname {Var} (W_{t})=\mathbb {E} (W_{t}^{2})-\mathbb {E} ^{2}(W_{t})=\mathbb {E} (W_{t}^{2})-0=\mathbb {E} (W_{t}^{2})=t.}
在维纳过程的独立增量定义中，令
t
2
=
t
{\displaystyle t_{2}=t}
s
2
=
t
1
=
s
<
t
{\displaystyle s_{2}=t_{1}=s<t}
s
1
=
0
{\displaystyle s_{1}=0}
W
s
=
W
t
1
−
W
s
1
∼
N
(
0
,
s
)
{\displaystyle W_{s}=W_{t_{1}}-W_{s_{1}}\sim {\mathcal {N}}(0,s)}
W
t
−
W
s
=
W
t
2
−
W
s
2
∼
N
(
0
,
t
−
s
)
{\displaystyle W_{t}-W_{s}=W_{t_{2}}-W_{s_{2}}\sim {\mathcal {N}}(0,t-s)}
cov
(
W
s
,
W
t
)
=
E
[
(
W
s
−
E
(
W
s
)
)
⋅
(
W
t
−
E
(
W
t
)
)
]
=
E
(
W
s
⋅
W
t
)
=
E
[
W
s
(
W
t
−
W
s
)
]
+
E
(
W
s
2
)
=
E
(
W
s
)
E
(
W
t
−
W
s
)
+
s
=
s
.
{\displaystyle \operatorname {cov} (W_{s},W_{t})=\mathbb {E} \left[(W_{s}-\mathbb {E} (W_{s}))\cdot (W_{t}-\mathbb {E} (W_{t}))\right]=\mathbb {E} (W_{s}\cdot W_{t})=\mathbb {E} [W_{s}\left(W_{t}-W_{s}\right)]+\mathbb {E} (W_{s}^{2})=\mathbb {E} (W_{s})\mathbb {E} \left(W_{t}-W_{s}\right)+s=s\ \ .}
所以两个不同时刻
0
⩽
s
,
t
{\displaystyle 0\leqslant s,t}
W
t
{\displaystyle W_{t}}
W
s
{\displaystyle W_{s}}
协方差 和相关系数 是：
cov
(
W
s
,
W
t
)
=
min
(
s
,
t
)
,
corr
(
W
s
,
W
t
)
=
c
o
v
(
W
s
,
W
t
)
σ
W
s
σ
W
t
=
min
(
s
,
t
)
s
t
=
min
(
s
,
t
)
max
(
s
,
t
)
.
{\displaystyle \operatorname {cov} (W_{s},W_{t})=\min(s,t)\,,\qquad \quad \operatorname {corr} (W_{s},W_{t})={\mathrm {cov} (W_{s},W_{t}) \over \sigma _{W_{s}}\sigma _{W_{t}}}={\frac {\min(s,t)}{\sqrt {st}}}={\sqrt {\frac {\min(s,t)}{\max(s,t)}}}\,.}
维纳过程中的即时最大值
M
t
=
max
0
≤
s
≤
t
W
s
{\displaystyle M_{t}=\max _{0\leq s\leq t}W_{s}}
W
t
{\displaystyle W_{t}}
f
M
t
,
W
t
(
m
,
w
)
=
2
(
2
m
−
w
)
t
2
π
t
e
−
(
2
m
−
w
)
2
2
t
,
m
≥
0
,
w
≤
m
{\displaystyle f_{M_{t},W_{t}}(m,w)={\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}},m\geq 0,w\leq m}
而即时最大值的分布
f
M
t
{\displaystyle f_{M_{t}}}
−
∞
<
w
≤
m
{\displaystyle -\infty <w\leq m}
f
M
t
(
m
)
=
∫
−
∞
m
f
M
t
,
W
t
(
m
,
w
)
d
w
=
∫
−
∞
m
2
(
2
m
−
w
)
t
2
π
t
e
−
(
2
m
−
w
)
2
2
t
d
w
=
2
π
t
e
−
m
2
2
t
{\displaystyle f_{M_{t}}(m)=\int _{-\infty }^{m}f_{M_{t},W_{t}}(m,w)\,dw=\int _{-\infty }^{m}{\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}}\,dw={\sqrt {\frac {2}{\pi t}}}e^{\frac {-m^{2}}{2t}}}
即时最大值的数学期望是[ 3] :114 ：
E
M
t
=
∫
0
∞
m
f
M
t
(
m
)
d
m
=
∫
0
∞
m
2
π
t
e
−
m
2
2
t
d
m
=
2
t
π
.
{\displaystyle \mathbb {E} M_{t}=\int _{0}^{\infty }mf_{M_{t}}(m)\,dm=\int _{0}^{\infty }m{\sqrt {\frac {2}{\pi t}}}e^{\frac {-m^{2}}{2t}}\,dm={\sqrt {\frac {2t}{\pi }}}.}
由于维纳过程上下对称，即时最小值显然是即时最大值的相反数 。
将一个维纳过程不断按比例展开，它的一部分就会呈现另一个维纳过程的样子 尺度不变性：对任意的正实数
α
>
0
{\displaystyle \alpha >0}
(
V
t
)
t
⩾
0
:
V
t
=
1
α
W
α
t
{\displaystyle \left(V_{t}\right)_{t\geqslant 0}:\,\,V_{t}={\frac {1}{\sqrt {\alpha }}}W_{\alpha t}}
时间反转：对任意的正实数
T
>
0
{\displaystyle T>0}
(
V
t
)
0
⩽
t
⩽
T
:
V
t
=
W
T
−
W
T
−
t
{\displaystyle \left(V_{t}\right)_{0\leqslant t\leqslant T}:\,\,V_{t}=W_{T}-W_{T-t}}
(
W
t
)
0
⩽
t
⩽
T
{\displaystyle \left(W_{t}\right)_{0\leqslant t\leqslant T}}
空间对称：随机过程
(
V
t
)
t
⩾
0
:
V
t
=
−
W
t
{\displaystyle \left(V_{t}\right)_{t\geqslant 0}:\,\,V_{t}=-W_{t}}
时间反演：随机过程
(
V
t
)
t
⩾
0
:
V
0
=
0
,
∀
t
>
0
,
V
t
=
t
W
1
t
{\displaystyle \left(V_{t}\right)_{t\geqslant 0}:\,\,V_{0}=0,\,\,\forall t>0,\,\,V_{t}=tW_{\frac {1}{t}}}
参考资料：[ 2] :13 、[ 4] :44
维纳过程具有马尔可夫性质 ，也就是说，在任意一点之后的走势仅仅和这一点的取值相关，而与之前的取值无关。也就是说，对任何的有界连续函数
ϕ
{\displaystyle \phi }
E
[
ϕ
(
W
s
,
s
⩾
t
)
|
F
t
]
=
E
[
ϕ
(
W
s
,
s
⩾
t
)
|
W
t
]
{\displaystyle \mathbb {E} [\phi (W_{s},s\geqslant t)|{\mathcal {F}}_{t}]=\mathbb {E} [\phi (W_{s},s\geqslant t)|W_{t}]}
因此维纳过程具有时间平移不变性：随机过程
(
V
t
)
t
⩾
0
:
V
t
=
W
t
0
+
t
−
W
t
0
{\displaystyle \left(V_{t}\right)_{t\geqslant 0}:\,\,V_{t}=W_{t_{0}+t}-W_{t_{0}}}
停时
τ
{\displaystyle \tau }
B
t
=
W
τ
+
t
−
W
τ
{\displaystyle B_{t}=W_{\tau +t}-W_{\tau }}
F
τ
{\displaystyle {\mathcal {F}}_{\tau }}
ϕ
{\displaystyle \phi }
E
[
ϕ
(
W
s
,
s
⩾
τ
)
|
F
τ
]
=
E
[
ϕ
(
W
s
,
s
⩾
τ
)
|
W
τ
]
.
{\displaystyle \mathbb {E} [\phi (W_{s},s\geqslant \tau )|{\mathcal {F}}_{\tau }]=\mathbb {E} [\phi (W_{s},s\geqslant \tau )|W_{\tau }].}
维纳过程的强马尔可夫性质，说明即便给定的时间不是定时而是一个停时，维纳过程在停时之后的走势仍然与之前无关。所以，将停时之后的维纳过程上下反转，仍然会是一个维纳过程。用数学语言来说，就是：给定一个停时
τ
{\displaystyle \tau }
B
t
=
W
t
1
t
⩽
τ
+
(
2
W
τ
−
W
t
)
1
t
>
τ
{\displaystyle B_{t}=W_{t}\mathbf {1} _{t\leqslant \tau }+\left(2W_{\tau }-W_{t}\right)\mathbf {1} _{t>\tau }}
作为推论，可以建立即时最大值
M
t
=
max
0
≤
s
≤
t
W
s
{\displaystyle M_{t}=\max _{0\leq s\leq t}W_{s}}
W
t
{\displaystyle W_{t}}
a
>
0
{\displaystyle a>0}
τ
a
=
inf
{
t
>
0
,
W
t
>
a
}
{\displaystyle \tau _{a}=\inf\{t>0,\,W_{t}>a\}}
{
τ
a
⩽
t
}
=
{
M
t
⩾
a
}
{\displaystyle \{\tau _{a}\leqslant t\}=\{M_{t}\geqslant a\}}
P
(
M
t
⩾
a
)
=
2
P
(
W
t
⩾
a
)
=
P
(
|
W
t
|
⩾
a
)
{\displaystyle \mathbb {P} \left(M_{t}\geqslant a\right)=2\mathbb {P} \left(W_{t}\geqslant a\right)=\mathbb {P} \left(|W_{t}|\geqslant a\right)}
a
>
b
⩾
0
{\displaystyle a>b\geqslant 0}
P
(
W
t
⩽
b
,
M
t
⩾
a
)
=
P
(
W
t
⩾
2
a
−
b
)
{\displaystyle \mathbb {P} \left(W_{t}\leqslant b,\,M_{t}\geqslant a\right)=\mathbb {P} \left(W_{t}\geqslant 2a-b\right)}
^ （英文） Rick Durrett. Probability: theory and examples ,4th edition. Cambridge University Press. 2000. ISBN 0521765390 ^ 2.0 2.1 2.2 （英文） Peter Mörters, Yuval Peres. Brownian Motion ISBN 9780521760188 ^ （英文） Steven E. Shreve. Stochastic Calculus for Finance II: Continuous Time Models ISBN 978-0-387-40101-0 ^ Nizar Touzi, Peter Tankov. Calcul Stochastique en Finance
Kleinert, Hagen , Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets , 4th edition, World Scientific (Singapore, 2004); Paperback ISBN 981-238-107-4 (also available online: PDF-files （页面存档备份 ，存于互联网档案馆 ）) Stark,Henry, John W. Woods , Probability and Random Processes with Applications to Signal Processing , 3rd edition, Prentice Hall (New Jersey, 2002); Textbook ISBN 0-13-020071-9
Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion , second edition, Springer-Verlag 1994.
![{\displaystyle \operatorname {cov} (W_{s},W_{t})=\mathbb {E} \left[(W_{s}-\mathbb {E} (W_{s}))\cdot (W_{t}-\mathbb {E} (W_{t}))\right]=\mathbb {E} (W_{s}\cdot W_{t})=\mathbb {E} [W_{s}\left(W_{t}-W_{s}\right)]+\mathbb {E} (W_{s}^{2})=\mathbb {E} (W_{s})\mathbb {E} \left(W_{t}-W_{s}\right)+s=s\ \ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f1c2b84b45a0c4d70b5de80940c73b81a3076e)
![{\displaystyle \mathbb {E} [\phi (W_{s},s\geqslant t)|{\mathcal {F}}_{t}]=\mathbb {E} [\phi (W_{s},s\geqslant t)|W_{t}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/243e1c79c424e37c7d1102b8443dab25d197d722)
![{\displaystyle \mathbb {E} [\phi (W_{s},s\geqslant \tau )|{\mathcal {F}}_{\tau }]=\mathbb {E} [\phi (W_{s},s\geqslant \tau )|W_{\tau }].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afbf5d6b9641b0f80186bd67cc73fe05b8f39691)
